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Properties

Label 1-2475-2475.1327-r0-0-0
Degree $1$
Conductor $2475$
Sign $-0.921 + 0.387i$
Analytic cond. $11.4938$
Root an. cond. $11.4938$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.994 + 0.104i)7-s + i·8-s + (−0.406 + 0.913i)13-s + (−0.913 − 0.406i)14-s + (−0.5 + 0.866i)16-s + (0.951 − 0.309i)17-s + 19-s + (0.207 + 0.978i)23-s + (−0.809 + 0.587i)26-s + (−0.587 − 0.809i)28-s + (−0.5 + 0.866i)29-s + (−0.104 + 0.994i)31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.994 + 0.104i)7-s + i·8-s + (−0.406 + 0.913i)13-s + (−0.913 − 0.406i)14-s + (−0.5 + 0.866i)16-s + (0.951 − 0.309i)17-s + 19-s + (0.207 + 0.978i)23-s + (−0.809 + 0.587i)26-s + (−0.587 − 0.809i)28-s + (−0.5 + 0.866i)29-s + (−0.104 + 0.994i)31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.921 + 0.387i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ -0.921 + 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3942915122 + 1.956880023i\)
\(L(\frac12)\) \(\approx\) \(0.3942915122 + 1.956880023i\)
\(L(1)\) \(\approx\) \(1.241793072 + 0.8072709390i\)
\(L(1)\) \(\approx\) \(1.241793072 + 0.8072709390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + T \)
23 \( 1 + (0.207 + 0.978i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.743 + 0.669i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.22430781454526526472914958981, −18.76845324710832221511127554310, −17.89598508697804795679753229522, −16.721605269943419491369483259736, −16.33843412469296231218621205707, −15.31611196168063984707565366279, −14.93693218280680646878896365537, −14.00361833179306326031667419955, −13.322372634192213653350530373496, −12.68085373849883537042092189994, −12.16921345216882046261780449459, −11.30452302219698559568739557884, −10.46432038397465035122253201833, −9.82556691716802592429955778666, −9.338650162714652704432537900430, −7.913338567568510086046554332285, −7.32229945972745795941102090693, −6.147902086639113713295109794451, −5.897668168016768222225539918343, −4.84845103872171605350693510177, −4.06808821663317458553281158548, −3.01833566221928200880975171642, −2.838077780257133717006493864712, −1.46571127706300802208908286435, −0.460744308439713666140875864694, 1.40703312489255667657594222570, 2.55784445210618653928573536634, 3.34738787502594423037756831776, 3.888516149337182559689216383108, 5.075603872927074232484453822602, 5.53782658091165839278714346473, 6.47026294574497689735706621861, 7.19160331562357168310846781751, 7.65253226974705446879472330111, 8.90841510607159798172075666166, 9.44356231462815445068223694292, 10.372030137441746241067032609308, 11.44837813411891811676293126934, 11.99907598145439713146717081222, 12.68378365148609467583300867564, 13.38130980852992737974788914415, 14.13725900612842233126725532459, 14.62614264277933431358526823991, 15.60424533689323504417687006879, 16.22624877995240378708283283498, 16.57991453023913438761279002194, 17.47574751046536081265929540935, 18.32976091037209510723784165751, 19.176360072436633175448523875274, 19.86642720254678033874882942917

Graph of the $Z$-function along the critical line